Watch This

Guidance

The Polar Coordinate System is alternative to the Cartesian Coordinate system you have used in the past to graph functions. The polar coordinate system is specialized for visualizing and manipulating angles.

Angles are identified by travelling counter-clockwise around the circular graph from the 0deg line, or
r-axis
(where the +
x
axis would be) to a specified angle.

To plot a specific point, first go along the
r
-axis by
r
units. Then, rotate counterclockwise by the given angle, commonly represented "θ". Be careful to use the correct units for the angle measure (either radians or degrees).

Radians

Usually polar plots are done with radians (especially if they include trigonometric functions), but sometimes degrees are used.

A
radian
is the angle formed between the
r
axis and a polar axis drawn to meet a section of the circumference that is the same length as the radius of a circle.

Given that the circumference of a circle is
, and since
r
is the radius, that means there are
radians
in a complete circle, and
radians
in 1/2 of a circle.

If 1/2 of a circle is
radians, and is 180deg, that means that there are
degrees in each
radian
.

That translates to approximately 57.3 degrees = 1 radian.

Graphing using technology

Polar Equations can be graphed using a graphing calculator: With the graphing calculator- go to
MODE
. There select
RADIAN
for the angle measure and
POL
(for Polar) on the
FUNC
(function)line. When Y = is pressed, note that the equation has changed from y = to r = . There input the polar equation. After pressing graph, if you can’t see the full graph, adjust
x
- and
y
- max/min, etc in
WINDOW.

Example A

Plot the points on a polar coordinate graph:

Point A

Point B

Point C

Solution

Below is the pole, polar axis and the points A, B and C.

Example B

Plot the following points:

a.

b.

c.

d.

e.

Solution

Example C

Use a graphing calculator or plotting program to plot the following equations:

a.

b.

Solution

a.

b.

Review the steps above under
Graphing using technology
if you are having trouble.

Vocabulary

The
polar coordinate system
is a specialized graph used for angles and angle manipulations.

The
pole
is the center point on a polar graph.

One
radian
is the angle formed by moving counter-clockwise around the circumference of a circle by the length of the radius. It is equal to apx 57.3 degrees.

The
polar axis
is a ray drawn from the pole at the 0
o
angle on a polar graph.

Guided Practice

1) Plot the points on a polar graph:

a)

b)

c)

2) Convert from radians to degrees:

a)

b)

c)

3) Convert from degrees to radians:

a)

b)

c)

4) Convert from degrees to radians, answer in terms of
:

a)

b)

c)

Answers

1) The points are plotted on the graph below:

2) Recall that
and

a) If
then

b) If
then

c) If
then

3) Recall that

a) If
then

b) If
then

c) If
then

4) Recall that
and therefore

a) If
then

b) If
and
then

c) If
then

Practice

Why can a point on the plane not be labeled using a unique ordered pair

Explain how to graph
if
and/or

Graph Each Point in the Polar Plane

A

B

C

D

E

Name Two Other Pairs of Polar Coordinates for Each Point

Graph Each Polar Equation

Find the Distance Between Points

and

and

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Polar Coordinates.